Theorem pwunss 4020

Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
pwunss	⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Proof of Theorem pwunss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun 3122	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵))
2		elun 3084	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵))
3		vex 2560	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	elpw 3365	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5	3	elpw 3365	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
6	4, 5	orbi12i 681	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵))
7	2, 6	bitri 173	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵))
8	3	elpw 3365	. . 3 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∪ 𝐵))
9	1, 7, 8	3imtr4i 190	. 2 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵))
10	9	ssriv 2949	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 629   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  𝒫 cpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  pwundifss  4022  pwunim  4023